Genericity of hyperbolicity and saddle-node bifurcations in reaction-diffusion equations depending on a parameter

Semilinear elliptic equations of the form ?ⁿ_i,j=1(a_ij(cursive Greek chi)u_{cursive Greek chii}(cursive Greek chi))_{cursive Greek chij} + f(?, cursive Greek chi, u(cursive Greek chi)) = 0, cursive ? O, u(cursive Greek chi) - 0, cursive Greek chi ? ?O, are considered, where ? ? R is a parameter, O ? Rⁿ is a bounded domain and f is a smooth non-linear function. It is shown that for 'generic' functions f, the set of non-trivial solutions (?, u) consists of a finite, or countable, collection of smooth, 1-dimensional curves and any such solution is either hyperbolic or is a saddle-node bifurcation point of the curve.